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PSAT Math : Geometry

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Example Questions

Example Question #4 : Sectors

Circle

In the circle above, the angle A in radians is $\frac{\pi}{4}$

What is the length of arc A?

Possible Answers:

$\frac{\pi r}{4}$

$\frac{\pi r}{16}$

$\frac{\pi r}{2}$

$\pi r$

$\frac{\pi r}{8}$

Correct answer:

$\frac{\pi r}{4}$

Explanation:

Circumference of a Circle = $2\pi r$

Arc Length

$=\frac{Angle A}{2\pi}\ast2\pi r$

$=\frac{\frac{\pi}{4}}{2\pi}\ast2\pi r$

$=\frac{\pi}{(4)2\pi}\ast2\pi r$

$=\frac{1}{8}\ast2\pi r=\frac{\pi r}{4}$

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Example Question #3 : Plane Geometry

Circle_120_degrees

If the area of a circle is $36\pi$ , then what is the length of the $120^\circ$ arc shown in the diagram?

Possible Answers:

$8\pi$

$4\pi$

$6\pi$

$12\pi$

Correct answer:

$4\pi$

Explanation:

We are given the area of the circle, but we need to find the circumference in order to find the arc length. The equation for the area of a circle is $A =\pi r^{2}$

Because we know that the area is 36 $\pi$ , we can use that equation to find the radius of the circle.

$36\pi =\pi r^{2}$

Divide both sides by $\pi$

$36 = r^{2}$

Take the square root of both sides, and see that the radius is 6.

We can now find the circumference of the circle using the formula $C=2\pi r$

$C=2\pi 6$

$C=12\pi$

Now that we know the circumference, we can set up a proportion. The length of the 120 degree arc is going to be only a portion of the total circumference of the circle. By putting the degree measure over 360 and setting it equal to x over the circumference, we can find exactly how long the arc is.

$\frac{120}{360}=\frac{x}{12\pi }$

When you multiply both sides by $12\pi$ , you find the solution:

$x=4\pi$

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Example Question #1 : Plane Geometry

A circular, 8-slice pizza is placed in a square box that has dimensions four inches larger than the diameter of the pizza. If the box covers a surface area of 256 in², what is the surface area of one piece of pizza?

Possible Answers:

18π in²

144π in²

4.5π in²

36π in²

9π in²

Correct answer:

4.5π in²

Explanation:

The first thing to do is calculate the dimensions of the pizza box. Based on our data, we know 256 = s². Solving for s (by taking the square root of both sides), we get 16 = s (or s = 16).

Now, we know that the diameter of the pizza is four inches less than 16 inches. That is, it is 12 inches. Be careful! The area of the circle is given in terms of radius, which is half the diameter, or 6 inches. Therefore, the area of the pizza is π * 6² = 36π in². If the pizza is 8-slices, one slice is equal to 1/8 of the total pizza or (36π)/8 = 4.5π in².

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Example Question #1 : How To Find The Area Of A Sector

If B is a circle with line AC = 12 and line BC = 16, then what is the area formed by DBE?

Possible Answers:

$200$

$256\pi$

$100\pi$

$144$

$5\pi$

Correct answer:

$100\pi$

Explanation:

Line AB is a radius of Circle B, which can be found using the Pythagorean Theorem:

$AB^2=AC^2+BC^2\rightarrow AB=\sqrt{AC^2+BC^2}=\sqrt{16^2+12^2}=\sqrt{400}=20$

Since AB is a radius of B, we can find the area of circle B via:

$Area=\pi R^2=\pi(20^2)=400\pi$

Angle DBE is a right angle, and therefore $\frac{1}{4}$ of the circle so it follows:

$Area(DBE)=\frac{400}{4}\pi=100\pi$

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Example Question #261 : Geometry

Circle

The radius of the circle above is and $\angle A=45^{\circ}$ . What is the area of the shaded section of the circle?

Possible Answers:

$2\pi$

$8\pi$

$16\pi$

$4\pi$

$\pi$

Correct answer:

$2\pi$

Explanation:

Area of Circle = πr² = π4²= 16π

Total degrees in a circle = 360

Therefore 45 degree slice = 45/360 fraction of circle = 1/8

Shaded Area = 1/8 * Total Area = 1/8 * 16π = 2π

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Example Question #2 : Sectors

Square-missing

SRAD is a square.

$\overline{AB}=\overline{BC}=\overline{CD}$

The arc from to is a semicircle with a center at the midpoint of $\overline{BC}$ .

All units are in feet.

The diagram shows a plot of land.

The cost of summer upkeep is $2.50 per square foot.

In dollars, what is the total upkeep cost for the summer?

Possible Answers:

$\$(2250-31.25\pi )$

$\$(900-25\pi )$

$\$(900-12.5\pi )$

$\$(2250-250\pi )$

$\$900$

Correct answer:

$\$(2250-31.25\pi )$

Explanation:

To solve this, we must begin by finding the area of the diagram, which is the area of the square less the area of the semicircle.

The area of the square is straightforward:

30 * 30 = 900 square feet

Because each side is 30 feet long, AB + BC + CD = 30.

We can substitute BC for AB and CD since all three lengths are the same:

BC + BC + BC = 30

3BC = 30

BC = 10

Therefore the diameter of the semicircle is 10 feet, so the radius is 5 feet.

The area of the semi-circle is half the area of a circle with radius 5. The area of the full circle is 5²π = 25π, so the area of the semi-circle is half of that, or 12.5π.

The total area of the plot is the square less the semicircle: 900 - 12.5π square feet

The cost of upkeep is therefore 2.5 * (900 – 12.5π) = $(2250 – 31.25π).

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Example Question #31 : Geometry

In the figure, PQ is the arc of a circle with center O. If the area of the sector is $3\pi$ what is the perimeter of sector?

Possible Answers:

$6 + \pi$

$12 + 2\pi$

$12 + \pi$

$3 + 2\pi$

$1 + \pi$

Correct answer:

$12 + \pi$

Explanation:

First, we figure out what fraction of the circle is contained in sector OPQ: $\frac{30^{\circ}}{360^{\circ}}= \frac{1}{12}$ , so the total area of the circle is $\dpi{100} \small 12\times 3\pi=36$ .

Using the formula for the area of a circle, ${\pi}r^{2}$ , we can see that $\dpi{100} \small r=6$ .

We can use this to solve for the circumference of the circle, $2{\pi}r$ , or $12{\pi}$ .

Now, OP and OQ are both equal to r, and PQ is equal to $\dpi{100} \small \frac{1}{12}$ of the circumference of the circle, or ${\pi}$ .

To get the perimeter, we add OP + OQ + PQ, which give us $12+{\pi}$ .

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Example Question #4 : How To Find The Area Of A Sector

Circle_120_degrees

What is the area of the $120^\circ$ sector above if the radius of circle is ?

Possible Answers:

$18\pi$

$6\pi$

$9\pi$

$81\pi$

$27\pi$

Correct answer:

$27\pi$

Explanation:

To find the area of a sector, first find the area of the whole circle.

The radius of the circle is 9, so

$Area =\pi9^{2}$ which can be reduced to $81\pi$ .

The area of the sector is only a portion of the total area.To find out exactly how large the area is, set up a proportion where one side equals the angle measure over 360

$\frac{120}{360}=\frac{x}{81\pi }$

Multiply both sides by 81 $\pi$ and you will solve for x, which equals $27\pi$

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Example Question #1 : Circles

Two pizzas are made to the same dimensions. The only difference is that Pizza 1 is cut into pieces at 30° angles and Pizza 2 is cut at 45° angles. They are sold by the piece, the first for $1.95 per slice and the second for $2.25 per slice. What is the difference in total revenue between Pizza 2 and Pizza 1?

Possible Answers:

–$2.70

–$5.40

$5.40

$2.70

Correct answer:

–$5.40

Explanation:

First, let's calculate how many slices there are per pizza. This is done by dividing 360° by the respective slice degrees:

Pizza 1: 360/30 = 12 slices

Pizza 2: 360/45 = 8 slices

Now, the total amount made per pizza is calculated by multiplying the number of slices by the respective cost per slice:

Pizza 1: 12 * 1.95 = $23.40

Pizza 2: 8 * 2.25 = $18.00

The difference between Pizza 2 and Pizza 1 is thus represented by: 18 – 23.40 = –$5.40

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Example Question #1 : How To Find The Angle Of A Sector

A pie has a diameter of 12". A piece is cut out, having a surface area of 4.5π. What is the angle of the cut?

Possible Answers:

25°

45°

90°

4.5°

12.5°

Correct answer:

45°

Explanation:

This is simply a matter of percentages. We first have to figure out what percentage of the surface area is represented by 4.5π. To do that, we must calculate the total surface area. If the diameter is 12, the radius is 6. Don't be tricked by this!

A = π * 6 * 6 = 36π

Now, 4.5π is 4.5π/36π percentage or 0.125 (= 12.5%)

To figure out the angle, we must take that percentage of 360°:

0.125 * 360 = 45°

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PSAT Math : Geometry

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #4 : Sectors

Example Question #3 : Plane Geometry

Example Question #1 : Plane Geometry

Example Question #1 : How To Find The Area Of A Sector

Example Question #261 : Geometry

Example Question #2 : Sectors

Example Question #31 : Geometry

Example Question #4 : How To Find The Area Of A Sector

Example Question #1 : Circles

Example Question #1 : How To Find The Angle Of A Sector

All PSAT Math Resources

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